Optimal Disturbances in Boundary Layers

Abstract

Streamwise streaks are ubiquitous in transitional boundary layers, particularly when subjected to freestream turbulence. Using the steady boundary-layer approximation, we numerically calculate the upstream disturbances experiencing maximum spatial energy growth. The calculations use techniques commonly employed when solving optimal-control problems for distributed parameter systems. The calculated optimal disturbances consist of streamwise vortices developing into streamwise streaks. The maximum possible energy growth was found to scale linearly with the distance from the leading edge. Based on this result, we propose a simple model for prediction of transition location. Available experiments have been used to correlate the single constant appearing in the model.

Numerical and experimental investigations of oblique boundary layer transition

“…As in the present study the observed streak growth in all these investigations agrees well with the theory of non-modal growth. Moreover, Andersson et al (1997) found good agreement with the experimental results by Westin et al (1994) when they investigated spatial non-modal growth.…”

“…In a physical experiment or a spatial simulation, however, disturbances grow in space. Recently non-modal growth in boundary layers has been considered in spatial formulations by Luchini (1996) and Andersson, Berggren & Henningson (1997. They found in their analyses that the maximum possible energy growth scales linearly with the distance from the leading edge.…”

Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations

“…In particular, Andersson et al (1998aAndersson et al ( , b, c, 1999a have calculated the optimal perturbations of a flat-plate boundary layer by a similar backtracking optimization technique, the main difference being that their energy ratio is defined as in (12) rather than in the Reynolds-independent form of (14), whereas their mathematical model is still constituted by the boundary-layer equations. They thus obtained steady amplification spectra in which dependence on the Reynolds number is accounted for in the definition of energy but not in the model equations.…”

Algebraic growth in a Blasius boundary layer: Nonlinear optimal disturbances

“…The problem of spatial stability was recently tackled by Luchini [13,14] and Andersson et al [15,16]. They found that the optimal initial disturbance is composed of stationary streamwise vortices whereas the induced velocity field is dominated by streamwise streaks.…”